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29
A posteriori error estimates for vertex centered finite volume approximations of convectiondiffusionreaction equations
 M2AN Math. Model. Numer. Anal
, 2000
"... This paper is devoted to the study of a posteriori error estimates for the scalar nonlinear convectiondiffusionreaction equation c t +r (uf(c)) r (Drc)+c = 0. The estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the L¹ ..."
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Cited by 39 (7 self)
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This paper is devoted to the study of a posteriori error estimates for the scalar nonlinear convectiondiffusionreaction equation c t +r (uf(c)) r (Drc)+c = 0. The estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the L¹norm, independent of the diffusion parameter D. The resulting a posteriori error estimate is used to define an grid adaptive solution algorithm for the finite volume scheme. Finally numerical
Finite Volume Methods: Foundation and Analysis
 ENCYCLOPEDIA OF COMPUTATIONAL MECHANICS, VOLUME 1, FUNDAMENTALS
, 2004
"... Finite volume methods are a class of discretization schemes that have proven highly successful in approximating the solution of a wide variety of conservation law systems. They are extensively used in fluid mechanics, meteorology, electromagnetics, semiconductor device simulation, models of biologi ..."
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Cited by 35 (1 self)
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Finite volume methods are a class of discretization schemes that have proven highly successful in approximating the solution of a wide variety of conservation law systems. They are extensively used in fluid mechanics, meteorology, electromagnetics, semiconductor device simulation, models of biological processes and many other engineering areas governed by conservative systems that can be written in integral control volume form. This article
Numerical methods for high dimensional HamiltonJacobi equations using radial basis functions
 JOURNAL OF COMPUTATIONAL PHYSICS
, 2004
"... We utilize radial basis functions to construct numerical schemes for HamiltonJacobi (HJ) equations on unstructured data sets in arbitrary dimensions. The computational setup is a meshless discretization of the physical domain. We derive monotone schemes on unstructured data sets to compute the visc ..."
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Cited by 30 (3 self)
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We utilize radial basis functions to construct numerical schemes for HamiltonJacobi (HJ) equations on unstructured data sets in arbitrary dimensions. The computational setup is a meshless discretization of the physical domain. We derive monotone schemes on unstructured data sets to compute the viscosity solutions. The Essentially NonOscillatory (ENO) mechanism is combined with radial basis function reconstruction to obtain high order schemes in the presence of gradient discontinuities. Numerical examples of time dependent HJ equations in 2, 3 and 4 dimensions illustrate the accuracy of the new methods.
A Posteriori Error Analysis And Adaptivity For Finite Element Approximations Of Hyperbolic Problems
, 1997
"... this article is to present an overview of recent developments in the area of a posteriori error estimation for finite element approximations of hyperbolic problems. The approach pursued here rests on the systematic use of hyperbolic duality arguments. We also discuss the question of computational ..."
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Cited by 25 (4 self)
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this article is to present an overview of recent developments in the area of a posteriori error estimation for finite element approximations of hyperbolic problems. The approach pursued here rests on the systematic use of hyperbolic duality arguments. We also discuss the question of computational implementation of the a posteriori error bounds into adaptive finite element algorithms
A Posteriori Error Estimate for Finite Volume Approximations of Convection Diffusion Problems
, 2002
"... We deduce an a posteriori error estimate for cell centered finite volume approximations of nonlinear degenerate parabolic equations. The error estimator is robust with respect to the diffusion coefficient (which may tend to 0) and is applicable either in the case of diffusion dominated or in the cas ..."
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Cited by 18 (5 self)
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We deduce an a posteriori error estimate for cell centered finite volume approximations of nonlinear degenerate parabolic equations. The error estimator is robust with respect to the diffusion coefficient (which may tend to 0) and is applicable either in the case of diffusion dominated or in the case of convection dominated solutions.
An Adaptive FiniteDifference Method for Traveltimes and Amplitudes
 GEOPHYSICS
, 1999
"... The point source traveltime field has an upwind singularity at the source point. Consequently, all formally highorder finitedifference eikonal solvers exhibit firstorder convergence and relatively large errors. Adaptive upwind finitedifference methods based on highorder Weighted Essentially NonO ..."
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Cited by 10 (1 self)
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The point source traveltime field has an upwind singularity at the source point. Consequently, all formally highorder finitedifference eikonal solvers exhibit firstorder convergence and relatively large errors. Adaptive upwind finitedifference methods based on highorder Weighted Essentially NonOscillatory (WENO) RungeKutta difference schemes for the paraxial eikonal equation overcome this difficulty. The method controls error by automatic grid refinement and coarsening based on an a posteriori error estimation. It achieves prescribed accuracy at far lower cost than does the fixedgrid method. Moreover, the achieved high accuracy of traveltimes yields reliable estimates of auxiliary quantities such as takeoff angles and geometrical spreading factors.
Continuous Dependence on the Nonlinearity of Viscosity Solutions of Parabolic Equations
"... This paper establishes an upper bound for u \Gamma v where u is a subsolution of u t + F (u; Dxu; D 2 x u) = 0; and v is a supersolution of v t +G(v;Dxv;D 2 x v) = 0; in (0; 1)\Theta\Omega with Neumannbondary conditions and where\Omega is an open convex set. 1. INTRODUCTION In this paper we ..."
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Cited by 9 (1 self)
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This paper establishes an upper bound for u \Gamma v where u is a subsolution of u t + F (u; Dxu; D 2 x u) = 0; and v is a supersolution of v t +G(v;Dxv;D 2 x v) = 0; in (0; 1)\Theta\Omega with Neumannbondary conditions and where\Omega is an open convex set. 1. INTRODUCTION In this paper we study the difference between a viscosity subsolution of the parabolic equation u t + F (u; D x u; D 2 x u) = 0; (1) and a viscosity supersolution of the equation v t +G(v;D x v; D 2 x v) = 0; (2) * Partially supported by the National Science Foundation (Grant DMS9806956). 1 2 COCKBURN, GRIPENBERG, AND LONDEN in (0; T ) \Theta\Omega when u(0; x) and v(0; x) are given. We take\Omega to be an open and convex subset of R d , and Neumann boundary conditions are imposed at the boundary points, if there are any. The fundamental monotonicity condition is that F and G are nondecreasing in their first and nonincreasing in their third variable. This will, for example, be the case for the equations u t \Gamma OE(D x u)\Deltau = 0; (3) and v t \Gamma fl(D x v)\Deltav = 0; (4) provided both OE and fl are nonnegative functions. But it turns out that for equations of this form we can obtain better results than those one obtains by considering them to be special cases of (1) and (2). For this reason we shall actually state our result for the equations u t + f(u; D x u; D 2 x u) \Gamma OE(D x u)\Deltau = 0; (5) and v t + g(v; D x v; D 2 x v) \Gamma fl(D x v)\Deltav = 0: (6) Obviously, equations (5) and (6) include, respectively, (1), (3) and (2), (4) as particular cases. The upper bound for u \Gamma v that we get involves the difference between the intial values, a function depending on the moduli of continuity of the initial values and a parameter ff, and the supremum ...
Adaptive Finite Volume Approximations for Weakly Coupled Convection Dominated Parabolic Systems
 IMA J. Numer. Anal
, 2002
"... We consider a class of implicit vertex centered nite volume schemes on unstructured grids to approximate solutions of weakly coupled nonlinear convectiondi usionreaction systems. An a posteriori error estimate is proven. The obtained L1error estimate is robust in the di usion coecient, i.e. it app ..."
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Cited by 7 (1 self)
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We consider a class of implicit vertex centered nite volume schemes on unstructured grids to approximate solutions of weakly coupled nonlinear convectiondi usionreaction systems. An a posteriori error estimate is proven. The obtained L1error estimate is robust in the di usion coecient, i.e. it applies in particular in the convection dominated case. Constructing a grid adaptive solution algorithm we present numerical experiments that underline the applicability of the theoretical results.
Two a posteriori error estimates for onedimensional scalar conservation laws
 SIAM J. Numer. Anal
"... Abstract. In this paper, we propose a posteriori local error estimates for numerical schemes in the context of onedimensional scalar conservation laws. We first consider the schemes for which a consistent incell entropy inequality can be derived. Then we extend this result to secondorder schemes ..."
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Cited by 7 (0 self)
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Abstract. In this paper, we propose a posteriori local error estimates for numerical schemes in the context of onedimensional scalar conservation laws. We first consider the schemes for which a consistent incell entropy inequality can be derived. Then we extend this result to secondorder schemes written in viscous form satisfying weak entropy inequalities. As an illustration, we show several numerical tests on the Burgers equation and we propose an adaptive algorithm for the selection of the mesh.
Finite volume relaxation schemes for multidimensional conservation laws
 Math. Comp
, 1997
"... �������� � We consider semidiscrete and fully discrete finite volume relaxation schemes for multidimensional scalar conservation laws. These schemes are constructed by appropriate discretization of a relaxation system and it is shown to converge to the entropy solution of the conservation law with a ..."
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Cited by 4 (0 self)
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�������� � We consider semidiscrete and fully discrete finite volume relaxation schemes for multidimensional scalar conservation laws. These schemes are constructed by appropriate discretization of a relaxation system and it is shown to converge to the entropy solution of the conservation law with a rate of h 1/4 in L ∞ ([0, T], L 1 loc (�d)). 1.